A Fractional Lie Group Method For Anomalous Diffusion Equations (1007.2488v5)

Abstract: Lie group method provides an efficient tool to solve a differential equation. This paper suggests a fractional partner for fractional partial differential equations using a fractional characteristic method. A space-time fractional diffusion equation is used as an example to illustrate the effectiveness of the Lie group method.

• The paper introduces a novel fractional Lie group method to systematically derive exact solutions for space-time fractional partial differential equations.
• It extends the classical method of characteristics via fractional Taylor series and employs the modified Riemann-Liouville derivative.
• The methodology classifies solutions based on Lie symmetries, enabling the iterative construction of multiple explicit solution forms for diffusion equations.

Fractional Lie Group Method for Diffusion Equations

This paper introduces a fractional Lie group method to solve space-time fractional diffusion equations, using a space-time fractional diffusion equation as an illustrative example. The approach leverages the modified Riemann-Liouville derivative and extends the classical Lie group method to fractional partial differential equations (FPDEs).

Fractional Derivatives and Characteristic Methods

The paper employs the modified Riemann-Liouville fractional derivative and presents essential properties of fractional calculus. It uses the following definition of fractional integration with respect to $(dx)^\alpha$:

$$_0 I_x^\alpha f(x) = \frac{1}{\Gamma (\alpha )}\int_0^x (x - \xi)^{\alpha - 1} f(\xi )d\xi = \frac{1}{\Gamma (\alpha +1)}\int_0^x f(\xi )(d\xi )^\alpha ,0 < \alpha \le 1$$

Additionally, the paper introduces a generalized fractional method of characteristics to solve linear space-time FPDEs of the form:

$$a(x,t)\frac{\partial ^{^\beta } u(x,t)}{\partial x^\beta } + b(x,t)\frac{\partial ^\alpha u(x,t)}{\partial t^\alpha } = c(x,t),0 < \alpha ,\beta \le 1$$

This method extends the traditional method of characteristics by using fractional Taylor series in two variables to derive generalized characteristic curves.

Application of Lie Group Method

The paper applies the Lie group method to the fractional diffusion equation:

$\frac{\partial ^\alpha u(x,t)}{\partial t^\alpha } = \frac{\partial ^{2\beta } u(x,t)}{\partial x^{2\beta } },0 < \alpha ,\;\beta \le 1,\;0 < x{\rm{,}\;0 < t$

It assumes a one-parameter Lie group of transformations and derives the fractional second-order prolongation. By setting the coefficients to zero, the paper obtains a set of linear fractional equations. The solution of these equations leads to the determination of infinitesimal generators and a basis for the Lie algebra, which includes a four-dimensional sub-algebra and one infinite-dimensional sub-algebra.

Classification of Solutions

The analysis results in a classification of solutions for the fractional diffusion equation. These solutions are derived using the proposed fractional method of characteristics and solving the symmetry equations. As an example, the paper examines a specific solution form:

$u^{\rm{(5)}= e^{\frac{t^\alpha \varepsilon ^2 }{\Gamma (1 + \alpha )} - \frac{x^{\beta} \varepsilon }{\Gamma (1 + \beta )} f(\frac{x^\beta }{\Gamma (1 + \beta )} - 2\varepsilon \frac{t^\alpha }{\Gamma (1 + \alpha )},~\frac{t^\alpha }{\Gamma (1 + \alpha )})$

The paper illustrates how new exact solutions can be generated through iterative manipulations, providing a method to construct a series of solutions $u_1, u_2, u_3 ... u_n$.

Conclusion

The paper introduces a fractional Lie group method for solving FPDEs, specifically focusing on the space-time fractional diffusion equation. The method uses the modified Riemann-Liouville derivative, extends the method of characteristics, and derives a classification of solutions based on Lie group symmetries. The approach offers a means to find exact solutions for fractional differential equations, addressing a gap in systematic solution methods. It also suggests further research into extending the Lie group method to fractional differential equations of fractional order 0 to 2.